Basins of Attraction, Long-Run Equilibria, and the Speed of Step-by-Step Evolution

Abstract

The paper provides a general analysis of the types of models with e-perturbations which have been used recently to discuss the evolution of social conventions. Two new measures of the size and structure of the basins of attraction of dynamic systems, the radius and coradius, are introduced in order to bound the speed with which evolution occurs. The main theorem uses these measures to provide a chacterization useful for determining long-run equilibria and rates of convergence. Evolutionary forces are most powerful when evolution may proceed via small steps through a series of intermediate steady states. A number of applications are discussed. The selection of the risk dominant equilibrium in 2x2 games is generalized to the seldction of 1/2-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long-run equilibria.

JEL Classification: C7